1.2 Counting primitive triangles 103
1.2 Counting primitive triangles ...........
We shall make use of the famous Euler polyhedron formula.
Theorem 1.2. If a closed polyhedron hasV vertices,Eedges andF
faces, thenV−E+F=2.
Given a lattice polygon with a partition into primitive lattice trian-
gles, we take two identical copies and glue them along their common
boundary. Imagine the 2-sheet polygon blown into a closed polyhedron.
The number of vertices isV =2I+B. Suppose there areTprimitive
triangles in each sheet. Then there areF=2Tfaces of the polyhedron.
Since every face is a triangle, and each edge is contained in exactly two
faces, we have 2 E=3F. It follows thatE=3T. Now, Euler’s poly-
hedron formula gives(2I+B)− 3 T+2T =2. From this, we have
T=2I+B− 2.